The ontological argument

I read your explanation of the ontological argument and didn't fully understand it.
If I understand correctly, it consists of two parts (chapters B and C in the Proslogion):
A. Something that exists is greater than something that does not exist. God can be conceived as existing. Therefore, God must be defined as existing, since otherwise something greater than him could be conceived.
on. Something that exists necessarily is greater than something that exists but not necessarily. God can be conceived as necessarily existing. Therefore God must be defined as necessarily existing, since otherwise something greater than him could be conceived.
Regarding claim A, I didn't understand why it proves that God exists in this world of ours. It only says that the only God who exists is God, since only He is the greatest thing imaginable. It doesn't say that the only God who exists in our world is God. (I think you yourself addressed this claim, and indeed you explained that the core of the proof is claim B).
Claim B does indeed prove that he exists in our world, since his existence is necessary, but I did not understand why the assumption that we can conceive of God as necessarily existing is correct. Necessary existence is not a title that we decide to give to whomever we want. It must stem precisely from logical necessity. In order to make this assumption about God, we need to find such a necessity, but if we bring such a proof, it will be proof of God's existence without the ontological argument.
 
Thank you in advance.